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IEEE International Symposium on Information Theory (ISIT)

Average spectra for ensembles of LDPC codes and applications

Irina E. Bocharova, Boris D. Kudryashov, Vitaly Skachek, Yauhen Yakimenka

Published Jun 1, 2017• 14 citations
Publisher

Abstract

The exact values of finite length average weight distributions for both binary ensembles and binary images of nonbinary ensembles of regular LDPC codes are computed. The exact average stopping set size distribution for the binary ensemble is also obtained. The computed spectra are applied in order to bound from above the average stopping redundancy of the ensemble of binary regular LDPC codes. The asymptotic typical normalized minimum distances for the binary image of the ensemble of nonbinary regular LDPC codes and the typical minimum stopping distances for the binary regular LDPC codes are also presented.

DOI
10.1109/isit.2017.8006550

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The exact values of finite length average weight distributions for both binary ensembles and binary images of nonbinary ensembles of regular LDPC codes are computed. The exact average stopping set size distribution for the binary ensemble is also obtained. The computed spectra are applied in order to bound from above the average stopping redundancy of the ensemble of binary regular LDPC codes. The asymptotic typical normalized minimum distances for the binary image of the ensemble of nonbinary regular LDPC codes and the typical minimum stopping distances for the binary regular LDPC codes are also presented.
The Gallager ensembles of binary regular LDPC codes and binary images of nonbinary regular LDPC codes are studied. Recurrent procedure for computing average spectra for these two ensembles is presented. By using the existing bounding techniques, estimates on the error probability of the maximum-likelihood (ML) decoding over an AWGN channel with BPSK signaling for short codes from different ensembles of LDPC codes are obtained. The numerical results show performance of the ML decoding for different code ensembles. Conclusions drawn based on the average code spectra are then verified by near-ML decoding simulations for both randomly selected and the best known short codes. The asymptotic ML decoding thresholds for AWGN and BSC channels are calculated. As expected, codes with the ML decoding performance superior to that of the average code in the ensemble, are easy to find. However, comparison of the the presented results with simulation results for belief propagation (BP) decoding shows that the ML decoding performance should not be used as a target for searching for good iteratively decodable codes.
In this paper, we study the asymptotic behavior of two quantities: the distribution of stopping sets, and the stopping number s/sup */-the size of the smallest nonempty stopping set in a code and use it to bound the error probability of LDPC codes with large blocklength. We derive an expression for the normalized average stopping set distribution of regular and irregular LDPC codes and use it to obtain a critical fraction of the block length above which codes have exponentially many stopping sets of that size.